On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique

Zoubida Bouazza; Sabit Souhila; Sina Etemad; Mohammed Said Souid; Ali Akgül; Shahram Rezapour; Manuel De la Sen; Zoubida Bouazza; Sabit Souhila; Sina Etemad; Mohammed Said Souid; Ali Akgül; Shahram Rezapour; Manuel De la Sen

doi:10.3934/math.2023276

AIMS Mathematics

2023, Volume 8, Issue 3: 5484-5501. doi: 10.3934/math.2023276

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On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique

1.
Department of Informatics, University of Tiaret, Tiaret, Algeria
2.
Laboratoire Matériaux et Structures, Departement of Mathematics, University of Tiaret, Tiaret, Algeria
3.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
4.
Department of Economic Sciences, University of Tiaret, Tiaret, Algeria
5.
Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Türkiye
6.
Mathematics Research Center, Department of Mathematics, Near East University, North Cyprus, Mersin 10, Nicosia 99138, Türkiye
7.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
8.
Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Leioa 48940, Bizkaia, Spain

Received: 16 October 2022 Revised: 03 December 2022 Accepted: 07 December 2022 Published: 19 December 2022
MSC : 34A12, 34A40, 47A10

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.
- fractional variable order,
- Caputo-Hadamard fractional derivative,
- upper-lower solutions,
- existence results
Citation: Zoubida Bouazza, Sabit Souhila, Sina Etemad, Mohammed Said Souid, Ali Akgül, Shahram Rezapour, Manuel De la Sen. On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique[J]. AIMS Mathematics, 2023, 8(3): 5484-5501. doi: 10.3934/math.2023276

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Abstract

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.

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